3.206 \(\int \frac {x^4}{-1+x^4} \, dx\)

Optimal. Leaf size=14 \[ x-\frac {1}{2} \tan ^{-1}(x)-\frac {1}{2} \tanh ^{-1}(x) \]

[Out]

x-1/2*arctan(x)-1/2*arctanh(x)

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {321, 212, 206, 203} \[ x-\frac {1}{2} \tan ^{-1}(x)-\frac {1}{2} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[x^4/(-1 + x^4),x]

[Out]

x - ArcTan[x]/2 - ArcTanh[x]/2

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {x^4}{-1+x^4} \, dx &=x+\int \frac {1}{-1+x^4} \, dx\\ &=x-\frac {1}{2} \int \frac {1}{1-x^2} \, dx-\frac {1}{2} \int \frac {1}{1+x^2} \, dx\\ &=x-\frac {1}{2} \tan ^{-1}(x)-\frac {1}{2} \tanh ^{-1}(x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 26, normalized size = 1.86 \[ x+\frac {1}{4} \log (1-x)-\frac {1}{4} \log (x+1)-\frac {1}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(-1 + x^4),x]

[Out]

x - ArcTan[x]/2 + Log[1 - x]/4 - Log[1 + x]/4

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fricas [A]  time = 0.41, size = 18, normalized size = 1.29 \[ x - \frac {1}{2} \, \arctan \relax (x) - \frac {1}{4} \, \log \left (x + 1\right ) + \frac {1}{4} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^4-1),x, algorithm="fricas")

[Out]

x - 1/2*arctan(x) - 1/4*log(x + 1) + 1/4*log(x - 1)

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giac [A]  time = 1.05, size = 20, normalized size = 1.43 \[ x - \frac {1}{2} \, \arctan \relax (x) - \frac {1}{4} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^4-1),x, algorithm="giac")

[Out]

x - 1/2*arctan(x) - 1/4*log(abs(x + 1)) + 1/4*log(abs(x - 1))

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maple [A]  time = 0.00, size = 19, normalized size = 1.36 \[ x -\frac {\arctan \relax (x )}{2}+\frac {\ln \left (x -1\right )}{4}-\frac {\ln \left (x +1\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(x^4-1),x)

[Out]

x+1/4*ln(x-1)-1/4*ln(x+1)-1/2*arctan(x)

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maxima [A]  time = 1.21, size = 18, normalized size = 1.29 \[ x - \frac {1}{2} \, \arctan \relax (x) - \frac {1}{4} \, \log \left (x + 1\right ) + \frac {1}{4} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^4-1),x, algorithm="maxima")

[Out]

x - 1/2*arctan(x) - 1/4*log(x + 1) + 1/4*log(x - 1)

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mupad [B]  time = 0.06, size = 10, normalized size = 0.71 \[ x-\frac {\mathrm {atan}\relax (x)}{2}-\frac {\mathrm {atanh}\relax (x)}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(x^4 - 1),x)

[Out]

x - atan(x)/2 - atanh(x)/2

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sympy [A]  time = 0.13, size = 19, normalized size = 1.36 \[ x + \frac {\log {\left (x - 1 \right )}}{4} - \frac {\log {\left (x + 1 \right )}}{4} - \frac {\operatorname {atan}{\relax (x )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(x**4-1),x)

[Out]

x + log(x - 1)/4 - log(x + 1)/4 - atan(x)/2

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